Problem 81 - Precalculus with Limits

Question

Condensing a Logarithmic Expression In Exercises \(67-82\) , condense the expression to the logarithm of a single quantity. $$\frac{1}{3}\left[\log _{8} y+2 \log _{8}(y+4)\right]-\log _{8}(y-1)$$

Step-by-Step Solution

Answer

\(\log _{8} \frac{(y^{1/3}(y+4)^2)}{(y-1)}\)

Step 1: Apply Logarithm Power Rule

In the original expression, \(\frac{1}{3}\left[\log _{8} y+2 \log _{8}(y+4)\right]-\log _{8}(y-1)\), apply the power rule of logarithm that allows moving back and forth the coefficients of the logarithm as powers of the argument. This moves the coefficients 1/3 and 2 at the positions of powers of the arguments y and (y+4) respectively, yielding \(\log _{8} y^{1/3}+\log _{8} (y+4)^2 - \log _{8}(y-1)\)

Step 2: Apply Logarithm Product Rule

Because the sum of two logarithms equals the logarithm of the product of two numbers (base being the same), the first two terms can be combined into one: \(\log _{8} (y^{1/3}(y+4)^2) - \log _{8}(y-1)\)

Step 3: Apply Logarithm Quotient Rule

The Quotient Rule of logarithms states that the logarithm of a quotient is the difference of the logarithms. Therefore the entire expression can finally be condensed to: \(\log _{8} \frac{(y^{1/3}(y+4)^2)}{(y-1)}\)

Key Concepts

Logarithm Power Rule

The logarithm power rule is a handy tool used when dealing with logarithms that have coefficients attached to them. This concept allows you to "move" a numerical coefficient in front of the logarithm to the exponent on the inside of the logarithm.
For example, if you have an expression like \(a \cdot \log_b(x)\), you can rewrite it as \( \log_b(x^a)\). Here, 'a' shifts from being just a multiplier to becoming an exponent of x inside the logarithm.
In the original problem, we applied the power rule by moving the coefficients \(\frac{1}{3}\) and \(2\) to the exponents of \(y\) and \((y+4)\), respectively. This transforms

• \(\frac{1}{3} \cdot \log _{8} y\) into \( \log _{8} y^{1/3}\)
• \(2 \cdot \log _{8}(y+4)\) into \( \log _{8} (y+4)^2\)

This rewrite effectively simplifies the logarithmic terms for further operations.

Logarithm Product Rule

The logarithm product rule is another essential component when simplifying logarithms. It states that the sum of two logarithms with the same base can be combined into a single logarithm, where the arguments are multiplied together.
This means that \( \log_b(x) + \log_b(y) = \log_b(x \cdot y)\).
In our exercise, after applying the power rule, we had two logarithmic expressions: \(\log _{8} y^{1/3}\) and \(\log _{8} (y+4)^2\). Using the product rule, these two expressions can be merged into:

• \(\log _{8} (y^{1/3}\cdot (y+4)^2)\)

This step helps consolidate our expression, making it simpler by reducing the number of terms involved.

Logarithm Quotient Rule

The last tool we use in simplifying our logarithmic expression is the logarithm quotient rule. This rule states that if you have the logarithm of a quotient, it is the same as the difference between two logarithms.
Mathematically, it looks like this: \(\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)\).
In the problem at hand, after using the power and product rules, we had arrive at \(\log _{8} (y^{1/3}(y+4)^2)\) and needed to subtract \(\log _{8}(y-1)\).
Applying the quotient rule, we condense these terms into a single expression:

• \(\log _{8} \frac{(y^{1/3}(y+4)^2)}{(y-1)}\)

The quotient rule simplifies logarithms by reducing subtraction between terms to division inside a single log, perfect for creating neat, condensed expressions.